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Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang Copyright ᮊ 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 7 ROBUST-OPTIMAL FUZZY CONTROL wx This chapter discusses the robust-optimal fuzzy control problem 1᎐3,which combines robust fuzzy control and optimal fuzzy control. The robust-optimal fuzzy control problem is useful for practical control system designs that call for both robustness and optimality. In the last two chapters the robustness and optimality issues have been addressed separately. This chapter presents a unified design procedure to address both issues simultaneously to provide a solution to the robust-optimal fuzzy control problem. A design example is included to illustrate the merits of robust fuzzy control, optimal fuzzy control, and robust-optimal fuzzy control. The well-known nonlinear control bench- mark problem, that is, the translational actuator with rotational actuator Ž. wx TORA system 4᎐6,is employed as the design example. 7.1 ROBUST-OPTIMAL FUZZY CONTROL PROBLEM The robust-optimal fuzzy control design conditions are captured in the following theorem. Naturally these conditions are rendered by combining Ž.Ž . Theorems 23 robust fuzzy control and 25 optimal fuzzy control . Ž. THEOREM 28 The PDC controller 2.23 that simultaneously considers both Ž. the robust fuzzy controller design Theorem 23 and the optimal fuzzy controller 121 ROBUST-OPTIMAL FUZZY CONTROL 122 Ž. design Theorem 25 can be designed by sol®ing the following LMIs: r 22 minimize ␭ q ␣␥ q ␤␥ Ä4 Ý iai i bi 22 ␭ , ␥ , ␥ , X , is1 ai bi M , ., M , Y 1 r 0 subject to X)0, Y G 0, 0 T ␭ x 0 Ž. ) 0,7.1 Ž. x 0 X Ž. ˆ S q s y 1 Y - 0, i s 1,2, .,r , Ž. ii 1 ˆ T y 2Y - 0, i - j F r s.t. h l h / ␾ , ij 2 ij ˆ U q s y 1 Y - 0, i s 1,2, .,r , Ž. ii 3 ˆ V y 2Y - 0, i - j F r s.t. h l h / ␾ , ij 4 ij where s ) 1, T XA q AX ii )) ) ) TT ž/ yBMy MB ii ii T D yI 00 0 ai ˆ S s , ii T D 0 yI 00 bi 2 EX 00y ␥ I 0 ai ai 2 yEM 00 0 y ␥ I bi i bi T XA q AX ii TT y BMy MB ij ji TTTTTT DDDD XE yME XE yME ai bi aj bj ai j bi aj i bj T q XA q AX jj 0 TT y BMy MB ji ij T D yI 000 0 0 0 0 ai T D 0 yI 00 0 0 0 0 bi ˆ T s , T ij D 00yI 00 0 0 0 aj T D 000yI 0000 bj 2 EX 0000y ␥ I 000 ai ai 2 y EM 0000 0 y ␥ I 00 bi j bi 2 EX 0000 0 0 y ␥ I 0 aj aj 2 y EM 0000 0 0 0 y ␥ I bj i bj ROBUST-OPTIMAL FUZZY CONTROL PROBLEM 123 Y 0000 Y s block-diag , Ž. 0 1 Y 00000000 Y s block-diag , Ž. 0 2 T XA q AX ii TT XC yM ii TT ž/ yBMy MB ii ii ˆ U s , ii y1 CX yW 0 i y1 yM 0 yR i T XA q AX ii TT yBMy MB ij ji TTTT XC yMXCyM ijji T qXA q AX jj 0 TT yBMy MB ji i j ˆ V s , ij y1 CX yW 000 i y1 yM 0 yR 00 j y1 CX 00yW 0 j y1 yM 000yR i Y 00 Y s block-diag , Ž. 0 3 Y 0000 Y s block-diag , Ž. 0 4 Ž. where the asterisk denotes the transposed elements matrices for symmetric positions. Proof. It follows directly from Theorems 23 and 25. Ž. Remark 20 As shown in Chapter 3, the condition 7.1 may be replaced with Ž. 3.56 to handle the uncertainty in initial conditions. Ž. When Q s 0 i.e., Y s XQ X , the relaxed conditions are reduced to the 000 following conditions: r 22 minimize ␭ q ␣␥ q ␤␥ Ä4 Ý iai i bi 22 ␭ , ␥ , ␥ , X , is1 ai bi M , ., M 1 r subject to X ) 0, ROBUST-OPTIMAL FUZZY CONTROL 124 T ␭ x 0 Ž. ) 0, x 0 X Ž. ˆ S - 0, i s 1,2, .,r , ii ˆ T - 0, i - j F r s.t. h l h / ␾ , ij i j ˆ U - 0, i s 1,2, .,r , ii ˆ V - 0, i - j F r s.t. h l h / ␾ . ij i j Ž. In the design problem above, the initial conditions x 0 are assumed known. If not so, the theorem is not directly applicable. In this case, if all the Ž. Ž. vertex points x 0 of a polyhedron containing the initial conditions x 0 are k known, that is, l x 0 s ␳ x 0, Ž. Ž. Ý kk k s1 l n ␳ G 0, ␳ s 1, x 0 g R , Ž. Ý kkk k s1 Theorem 28 can be modified as follows to handle the uncertain initial conditions. Ž. THEOREM 29 The PDC controller 2.23 that simultaneously considers both Ž. the robust fuzzy controller design Theorem 23 and the optimal fuzzy control Ž. design Theorem 25 can be designed by sol®ing the following LMIs: r 22 minimize ␭ q ␣␥ q ␤␥ Ä4 Ý iai i bi 22 ␭ , ␥ , ␥ , X , is1 ai bi M , ., M , Y 1 r 0 subject to X)0 Y G 0, 0 T ␭ x 0 Ž. k ) 0, k s 1,2, .,l, x 0 X Ž. k ˆ S q s y 1 Y - 0, i s 1,2, .,r Ž. ii 1 ˆ T y 2Y - 0, i - j F r s.t. h l h / ␾ ij 2 ij ˆ U q s y 1 Y - 0, i s 1,2, .,r Ž. ii 3 ˆ V y 2Y - 0. i - j F r s.t. h l h / ␾ ij 4 ij Proof. It follows directly from Theorem 28. DESIGN EXAMPLE: TORA 125 Fig. 7.1 TORA system. 7.2 DESIGN EXAMPLE: TORA Consider the system shown in Figure 7.1, which represents a translational Ž.wx oscillator with an eccentric rotational proof mass actuator TORA 4᎐6. The nonlinear coupling between the rotational motion of the actuator and the translational motion of the oscillator provides the mechanism for control. Let x and x denote the translational position and velocity of the cart 12 with x s x . Let x s ␪ and x s x denote the angular position and ˙˙ 21 3 43 velocity of the rotational proof mass. Then the system dynamics can be described by the equation x s f x q g x u q d,7.2 Ž. Ž. Ž . ˙ where u is the torque applied to the eccentric mass, d is the disturbance, and x 2 2 yx q ␧ x sin x 143 22 1 y ␧ cos x 3 f x s , Ž. x 4 2 ␧ cos xxy ␧ x sin x Ž. 31 4 3 22 1 y ␧ cos x 3 0 y ␧ cos x 3 22 1 y ␧ cos x 3 g x s , Ž. 0 1 22 1 y ␧ cos x 3 ␧ s 0.1. ROBUST-OPTIMAL FUZZY CONTROL 126 wx Consider the case of no disturbance, as in 4᎐6 , introduce new state variables z s x q ␧ sin x , z s x q ␧ x cos x , y s x , y s x , and em- 11 322 4 31324 ploy the feedback transformation 1 2 ␯ s ␧ cos yzy 1 q y ␧ sin y q u Ž. Ž. 11 2 1 22 1 y ␧ cos y 1 s ␣ z , y q ␤ yu Ž.Ž. 11 1 to bring the system into the following form: z s z ,7.3 Ž. ˙ 12 z syz q ␧ sin y ,7.4 Ž. ˙ 21 1 y s y ,7.5 Ž. ˙ 12 y s ␯ .7.6 Ž. ˙ 2 Ž. w 0 x 0 The equilibrium point of system 7.2 can be any point 0, 0, x , 0 , where x 33 wx is an arbitrary constant. Consider 0, 0, 0, 0 as the desired equilibrium point. The linearization around this point has a pair of nonzero imaginary eigenval- Ž. ues and two zero eigenvalues. Hence the system 7.2 at the origin is an example of a critical nonlinear system. This control problem is interpreted as a regulator problem of z ™ 0, z ™ 0, y ™ 0, and y ™ 0. 121 2 Ž.Ž. The T-S model of the TORA system can be constructed from 7.3 ᎐ 7.6 by using the fuzzy model construction described in Chapter 2: Rule 1 Ž. IF ytis‘‘about y ␲ or ␲ rad,’’ 1 THEN x t s Axt q B ut, Ž. Ž. Ž. ˙ 11 y t s Cxt . Ž. Ž. 1 Rule 2 ␲ ␲ Ž. IF ytis ‘‘about y or rad,’’ 1 22 THEN x t s Axt q B ut, Ž. Ž. Ž. ˙ 22 y t s Cxt . Ž. Ž. 2 DESIGN EXAMPLE: TORA 127 Rule 3 Ž. Ž. IF ytis ‘‘about 0 rad’’ and ytis ‘‘about 0,’’ 12 THEN x t s Axt q B ut, Ž. Ž. Ž. ˙ 33 y t s Cxt . Ž. Ž. 3 Rule 4 Ž. Ž. IF ytis ‘‘about 0 rad’’ and ytis ‘‘about ya or a,’’ 12 THEN x t s Axt q B ut, Ž. Ž. Ž. ˙ 44 y t s Cxt , Ž. Ž. 4 T Ž. w Ž. Ž. Ž. Ž.x Here, x t s zt, zt, yt, yt, 12 12 0100 0 Ž. sin ␣␲ 0 y10 ␧ 0 ␣␲ A s , B s , 0 11 0001 1 y ␧ 2 000 1 y ␧ 2 1 y ␧ 0100 0 2 y10 ␧ 0 0 ␲ A s , B s , 22 0 0001 1 0000 0100 0 y10 ␧ 0 0 0001 A s , B s , 0 33 2 1 ␧ y ␧ 00 2 22 1 y ␧ 1 y ␧ 1 y ␧ 0100 0 y10 ␧ 0 0 0001 A s , B s , 0 44 22 1 Ž. ␧ y ␧ 1 q a 00 2 22 1 y ␧ 1 y ␧ 1 y ␧ ROBUST-OPTIMAL FUZZY CONTROL 128 1000 0100 C s C s C s C s . 1234 0010 0001 wxŽ. In this simulation, x gya, aas 4 and 0 - ␣ - 1 instead of ␣ s 1 4 Ž. e.g., ␣ s 0.99 is used to maintain the controllability of the subsystem Ž. A , B in Rule 1. 11 The above fuzzy model is represented as r x t s h z t Ax t q Bu t ,7.7 Ä4 Ž. Ž. Ž. Ž. Ž . Ž. ˙ Ý iii i s1 r y t s h z t Cx t ,7.8 Ž. Ž. Ž. Ž . Ž. Ý ii i s1 Ž. w Ž. Ž.x ŽŽ where r s 4 and z t s yt yt. Here, h z t is the weight of the ith 12 i rules calculated by the membership values. Figure 7.2 shows the membership functions. The PDC fuzzy controller is designed as follows: Control Rule 1 Ž. IF ytis ‘‘about y ␲ or ␲ rad,’’ 1 Ž. Ž. THEN u t syFxt . 1 Control Rule 2 ␲ ␲ Ž. IF ytis ‘‘about y or rad,’’ 1 22 Ž. Ž. THEN u t syFxt . 2 Fig. 7.2 Membership functions. DESIGN EXAMPLE: TORA 129 Control Rule 3 Ž. Ž. IF ytis ‘‘about 0 rad’’ and ytis ‘‘about 0,’’ 12 Ž. Ž. THEN u t syFxt . 3 Control Rule 4 Ž. Ž. IF ytis ‘‘about 0 rad’’ and ytis ‘‘about ya or a,’’ 12 Ž. Ž. THEN u t syFxt . 4 w Figure 7.3 shows the comparison between a stable fuzzy controller satisfy- Ž. Ž.x Ž ing 3.23 and 3.24 and a robust fuzzy controller satisfying the conditions . in Theorem 23 for the TORA system with parameter change ␧ s 0.05. Figure 7.4 compares the performance of the stable fuzzy controller and an Ž. optimal fuzzy controller satisfying the conditions in Theorem 25 for the nominal TORA system. Figure 7.5 shows the control results of the robust Ž. Fig. 7.3 Control results for TORA with parameter change ␧ s 0.05 . Fig. 7.4 Control results for the nominal TORA. ROBUST-OPTIMAL FUZZY CONTROL 130 Ž. Fig. 7.5 Control results for TORA with parameter change ␧ s 0.05 . Fig. 7.6 Control results for the nominal TORA. Ž fuzzy controller and the robust-optimal fuzzy controller satisfying the condi- . tions in Theorem 28 for the TORA with the parameter change. Figure 7.6 compares the control results of the optimal fuzzy controller and the robust- optimal fuzzy controller for the nominal TORA. In all cases, the fuzzy control designs get the job done but with different performance characteris- tics. The robust-optimal fuzzy controller is the most versatile in that it addresses both the robustness and the optimality. REFERENCES 1. K. Tanaka, T. Taniguchi, and H. O. Wang, ‘‘Robust and Optimal Fuzzy Control: A Linear Matrix Inequality Approach,’’ 1999 International Federation of Automatic Ž. Control IFAC World Congress, Beijing, July 1999, pp. 213᎐218. [...]...REFERENCES 131 2 K Tanaka, M Nishimura, and H O Wang, ‘‘Multi-objective Fuzzy Control of High RiserHigh Speed Elevators using LMIs,’’ 1998 American Control Conference, 1998, pp 3450᎐3454 3 K Tanaka, T Taniguchi, and H O Wang, ‘‘Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs that Represent Decay Rate, . Ž. Ž . ISBNs: 0-4 7 1-3 232 4-1 Hardback ; 0-4 7 1-2 245 9-6 Electronic CHAPTER 7 ROBUST-OPTIMAL FUZZY CONTROL wx This chapter discusses the robust-optimal fuzzy. s y 1 Y - 0, i s 1,2, .,r , Ž. ii 1 ˆ T y 2Y - 0, i - j F r s.t. h l h / ␾ , ij 2 ij ˆ U q s y 1 Y - 0, i s 1,2, .,r , Ž. ii 3 ˆ V y 2Y - 0, i - j F r